Properties

Label 3630.2.a.n.1.1
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} -1.00000 q^{20} +4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -8.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -8.00000 q^{43} -1.00000 q^{45} +4.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +8.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} +6.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -12.0000 q^{67} +2.00000 q^{68} -4.00000 q^{69} -12.0000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -8.00000 q^{76} +2.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} -2.00000 q^{85} -8.00000 q^{86} +2.00000 q^{87} -6.00000 q^{89} -1.00000 q^{90} +4.00000 q^{92} -8.00000 q^{93} -4.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} -7.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −8.00000 −1.29777
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 4.00000 0.589768
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 1.00000 0.129099
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −8.00000 −0.862662
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) −8.00000 −0.829561
\(94\) −4.00000 −0.412568
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −7.00000 −0.707107
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −2.00000 −0.198030
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 8.00000 0.749269
\(115\) −4.00000 −0.373002
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) 6.00000 0.543214
\(123\) 6.00000 0.541002
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 2.00000 0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 1.00000 0.0860663
\(136\) 2.00000 0.171499
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −4.00000 −0.340503
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) −2.00000 −0.165521
\(147\) 7.00000 0.577350
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −8.00000 −0.648886
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 2.00000 0.160128
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) −8.00000 −0.611775
\(172\) −8.00000 −0.609994
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 4.00000 0.294884
\(185\) 2.00000 0.147043
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) −14.0000 −1.00514
\(195\) −2.00000 −0.143223
\(196\) −7.00000 −0.500000
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 1.00000 0.0707107
\(201\) 12.0000 0.846415
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 6.00000 0.419058
\(206\) 16.0000 1.11477
\(207\) 4.00000 0.278019
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 2.00000 0.137361
\(213\) 12.0000 0.822226
\(214\) −20.0000 −1.36717
\(215\) 8.00000 0.545595
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 2.00000 0.134231
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 8.00000 0.529813
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −2.00000 −0.130744
\(235\) 4.00000 0.260931
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 1.00000 0.0645497
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 6.00000 0.384111
\(245\) 7.00000 0.447214
\(246\) 6.00000 0.382546
\(247\) 16.0000 1.01806
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) −1.00000 −0.0632456
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) −2.00000 −0.123797
\(262\) −12.0000 −0.741362
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −12.0000 −0.733017
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 1.00000 0.0608581
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −16.0000 −0.959616
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 4.00000 0.238197
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) −12.0000 −0.712069
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 14.0000 0.820695
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 7.00000 0.408248
\(295\) −4.00000 −0.232889
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) −8.00000 −0.462652
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) −6.00000 −0.344691
\(304\) −8.00000 −0.458831
\(305\) −6.00000 −0.343559
\(306\) 2.00000 0.114332
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) −8.00000 −0.454369
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 2.00000 0.113228
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −2.00000 −0.112154
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 20.0000 1.11629
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 20.0000 1.10770
\(327\) −6.00000 −0.331801
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −4.00000 −0.219529
\(333\) −2.00000 −0.109599
\(334\) −16.0000 −0.875481
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) −9.00000 −0.489535
\(339\) −2.00000 −0.108625
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 4.00000 0.215353
\(346\) −18.0000 −0.967686
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 2.00000 0.107211
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) −4.00000 −0.212598
\(355\) 12.0000 0.636894
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 45.0000 2.36842
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) −6.00000 −0.313625
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 4.00000 0.208514
\(369\) −6.00000 −0.312348
\(370\) 2.00000 0.103975
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −4.00000 −0.206284
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 36.0000 1.84920 0.924598 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 8.00000 0.410391
\(381\) −16.0000 −0.819705
\(382\) −12.0000 −0.613973
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) −8.00000 −0.406663
\(388\) −14.0000 −0.710742
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) −2.00000 −0.101274
\(391\) 8.00000 0.404577
\(392\) −7.00000 −0.353553
\(393\) 12.0000 0.605320
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −24.0000 −1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 12.0000 0.598506
\(403\) −16.0000 −0.797017
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) −2.00000 −0.0990148
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 6.00000 0.296319
\(411\) −2.00000 −0.0986527
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 4.00000 0.196352
\(416\) −2.00000 −0.0980581
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 2.00000 0.0971286
\(425\) 2.00000 0.0970143
\(426\) 12.0000 0.581402
\(427\) 0 0
\(428\) −20.0000 −0.966736
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 6.00000 0.287348
\(437\) −32.0000 −1.53077
\(438\) 2.00000 0.0955637
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) −4.00000 −0.190261
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 2.00000 0.0949158
\(445\) 6.00000 0.284427
\(446\) −16.0000 −0.757622
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) −8.00000 −0.375873
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 14.0000 0.654177
\(459\) −2.00000 −0.0933520
\(460\) −4.00000 −0.186501
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 8.00000 0.370991
\(466\) 18.0000 0.833834
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 4.00000 0.184506
\(471\) 2.00000 0.0921551
\(472\) 4.00000 0.184115
\(473\) 0 0
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) −16.0000 −0.731823
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 1.00000 0.0456435
\(481\) 4.00000 0.182384
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 0 0
\(485\) 14.0000 0.635707
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 6.00000 0.271607
\(489\) −20.0000 −0.904431
\(490\) 7.00000 0.316228
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 6.00000 0.270501
\(493\) −4.00000 −0.180151
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 16.0000 0.714827
\(502\) −20.0000 −0.892644
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 16.0000 0.709885
\(509\) −22.0000 −0.975133 −0.487566 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 2.00000 0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) 26.0000 1.14681
\(515\) −16.0000 −0.705044
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 2.00000 0.0877058
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −2.00000 −0.0868744
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 6.00000 0.259645
\(535\) 20.0000 0.864675
\(536\) −12.0000 −0.518321
\(537\) 12.0000 0.517838
\(538\) −22.0000 −0.948487
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 2.00000 0.0857493
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 2.00000 0.0854358
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) −2.00000 −0.0848953
\(556\) −16.0000 −0.678551
\(557\) 46.0000 1.94908 0.974541 0.224208i \(-0.0719796\pi\)
0.974541 + 0.224208i \(0.0719796\pi\)
\(558\) 8.00000 0.338667
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 4.00000 0.168430
\(565\) −2.00000 −0.0841406
\(566\) 8.00000 0.336265
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) −8.00000 −0.335083
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −13.0000 −0.540729
\(579\) −22.0000 −0.914289
\(580\) 2.00000 0.0830455
\(581\) 0 0
\(582\) 14.0000 0.580319
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 2.00000 0.0826898
\(586\) 6.00000 0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 7.00000 0.288675
\(589\) −64.0000 −2.63707
\(590\) −4.00000 −0.164677
\(591\) 10.0000 0.411345
\(592\) −2.00000 −0.0821995
\(593\) 26.0000 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 24.0000 0.982255
\(598\) −8.00000 −0.327144
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) 8.00000 0.323645
\(612\) 2.00000 0.0808452
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) −16.0000 −0.643614
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) −8.00000 −0.321288
\(621\) −4.00000 −0.160514
\(622\) −20.0000 −0.801927
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) −16.0000 −0.634941
\(636\) −2.00000 −0.0793052
\(637\) 14.0000 0.554700
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) −1.00000 −0.0395285
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 20.0000 0.789337
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) −16.0000 −0.629512
\(647\) −44.0000 −1.72982 −0.864909 0.501928i \(-0.832624\pi\)
−0.864909 + 0.501928i \(0.832624\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) −6.00000 −0.234619
\(655\) 12.0000 0.468879
\(656\) −6.00000 −0.234261
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −12.0000 −0.466393
\(663\) 4.00000 0.155347
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −8.00000 −0.309761
\(668\) −16.0000 −0.619059
\(669\) 16.0000 0.618596
\(670\) 12.0000 0.463600
\(671\) 0 0
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) −10.0000 −0.385186
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −8.00000 −0.305888
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) −8.00000 −0.304997
\(689\) −4.00000 −0.152388
\(690\) 4.00000 0.152277
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 16.0000 0.606915
\(696\) 2.00000 0.0758098
\(697\) −12.0000 −0.454532
\(698\) 30.0000 1.13552
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 46.0000 1.73740 0.868698 0.495342i \(-0.164957\pi\)
0.868698 + 0.495342i \(0.164957\pi\)
\(702\) 2.00000 0.0754851
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) −4.00000 −0.150649
\(706\) 26.0000 0.978523
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 12.0000 0.450352
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 16.0000 0.597531
\(718\) 32.0000 1.19423
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 45.0000 1.67473
\(723\) 18.0000 0.669427
\(724\) −2.00000 −0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) −16.0000 −0.591781
\(732\) −6.00000 −0.221766
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 16.0000 0.590571
\(735\) −7.00000 −0.258199
\(736\) 4.00000 0.147442
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) 2.00000 0.0735215
\(741\) −16.0000 −0.587775
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) −8.00000 −0.293294
\(745\) 10.0000 0.366372
\(746\) −2.00000 −0.0732252
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −4.00000 −0.145865
\(753\) 20.0000 0.728841
\(754\) 4.00000 0.145671
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 36.0000 1.30758
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −16.0000 −0.579619
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) −2.00000 −0.0723102
\(766\) 12.0000 0.433578
\(767\) −8.00000 −0.288863
\(768\) −1.00000 −0.0360844
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) −26.0000 −0.936367
\(772\) 22.0000 0.791797
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) −8.00000 −0.287554
\(775\) 8.00000 0.287368
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) −22.0000 −0.788738
\(779\) 48.0000 1.71978
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 2.00000 0.0714742
\(784\) −7.00000 −0.250000
\(785\) 2.00000 0.0713831
\(786\) 12.0000 0.428026
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −10.0000 −0.356235
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) −2.00000 −0.0709773
\(795\) 2.00000 0.0709327
\(796\) −24.0000 −0.850657
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 22.0000 0.774437
\(808\) 6.00000 0.211079
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −20.0000 −0.700569
\(816\) −2.00000 −0.0700140
\(817\) 64.0000 2.23908
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 4.00000 0.139010
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 4.00000 0.138842
\(831\) −14.0000 −0.485655
\(832\) −2.00000 −0.0693375
\(833\) −14.0000 −0.485071
\(834\) 16.0000 0.554035
\(835\) 16.0000 0.553703
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 4.00000 0.138178
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 38.0000 1.30957
\(843\) −10.0000 −0.344418
\(844\) 0 0
\(845\) 9.00000 0.309609
\(846\) −4.00000 −0.137523
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) −8.00000 −0.274559
\(850\) 2.00000 0.0685994
\(851\) −8.00000 −0.274236
\(852\) 12.0000 0.411113
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) −20.0000 −0.683586
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 18.0000 0.612018
\(866\) 2.00000 0.0679628
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) −2.00000 −0.0678064
\(871\) 24.0000 0.813209
\(872\) 6.00000 0.203186
\(873\) −14.0000 −0.473828
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 16.0000 0.539974
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 58.0000 1.95407 0.977035 0.213080i \(-0.0683494\pi\)
0.977035 + 0.213080i \(0.0683494\pi\)
\(882\) −7.00000 −0.235702
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −4.00000 −0.134535
\(885\) 4.00000 0.134459
\(886\) −28.0000 −0.940678
\(887\) −40.0000 −1.34307 −0.671534 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 32.0000 1.07084
\(894\) 10.0000 0.334450
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 10.0000 0.333704
\(899\) −16.0000 −0.533630
\(900\) 1.00000 0.0333333
\(901\) 4.00000 0.133259
\(902\) 0 0
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 2.00000 0.0664822
\(906\) −8.00000 −0.265782
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −12.0000 −0.398234
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) 38.0000 1.25693
\(915\) 6.00000 0.198354
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 8.00000 0.262896
\(927\) 16.0000 0.525509
\(928\) −2.00000 −0.0656532
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 8.00000 0.262330
\(931\) 56.0000 1.83533
\(932\) 18.0000 0.589610
\(933\) 20.0000 0.654771
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 54.0000 1.76410 0.882052 0.471153i \(-0.156162\pi\)
0.882052 + 0.471153i \(0.156162\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 4.00000 0.130466
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 2.00000 0.0651635
\(943\) −24.0000 −0.781548
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −8.00000 −0.259554
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −38.0000 −1.23094 −0.615470 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 2.00000 0.0647524
\(955\) 12.0000 0.388311
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) 4.00000 0.128965
\(963\) −20.0000 −0.644491
\(964\) −18.0000 −0.579741
\(965\) −22.0000 −0.708205
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) 16.0000 0.513994
\(970\) 14.0000 0.449513
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 2.00000 0.0640513
\(976\) 6.00000 0.192055
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) −20.0000 −0.639529
\(979\) 0 0
\(980\) 7.00000 0.223607
\(981\) 6.00000 0.191565
\(982\) 12.0000 0.382935
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) 6.00000 0.191273
\(985\) 10.0000 0.318626
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 8.00000 0.254000
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) 4.00000 0.126745
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 4.00000 0.126618
\(999\) 2.00000 0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3630.2.a.n.1.1 1
11.10 odd 2 330.2.a.a.1.1 1
33.32 even 2 990.2.a.j.1.1 1
44.43 even 2 2640.2.a.n.1.1 1
55.32 even 4 1650.2.c.l.199.1 2
55.43 even 4 1650.2.c.l.199.2 2
55.54 odd 2 1650.2.a.r.1.1 1
132.131 odd 2 7920.2.a.bb.1.1 1
165.32 odd 4 4950.2.c.g.199.2 2
165.98 odd 4 4950.2.c.g.199.1 2
165.164 even 2 4950.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
330.2.a.a.1.1 1 11.10 odd 2
990.2.a.j.1.1 1 33.32 even 2
1650.2.a.r.1.1 1 55.54 odd 2
1650.2.c.l.199.1 2 55.32 even 4
1650.2.c.l.199.2 2 55.43 even 4
2640.2.a.n.1.1 1 44.43 even 2
3630.2.a.n.1.1 1 1.1 even 1 trivial
4950.2.a.k.1.1 1 165.164 even 2
4950.2.c.g.199.1 2 165.98 odd 4
4950.2.c.g.199.2 2 165.32 odd 4
7920.2.a.bb.1.1 1 132.131 odd 2